obtained from the equations in (REF ) via compactness and a partition of unity. Thus it suffices to prove the local graph representations in (REF ). The expansion for \(\varphi _{ode,\varepsilon }(x)\)  follows directly from [1]}, and the expansion for \(\varphi _{\varepsilon ,h}(x)\)  up to and including \(O(\varepsilon )\)  follows directly from Theorem REF . Carrying out the matching argument in the proof of Proposition REF  for the map (REF ) and expanding in powers of \(\varepsilon h\)  instead of \(\varepsilon \)  leads to the higher order correction \(O(\varepsilon ^2 h)\)  (a factor of \(h\)  cancels out since \(h\)  is a common factor of \(\bar{z} - z\) ).
